Answer to Question #105106 in Abstract Algebra for Sourav Mondal

Question #105106
Consider the ideal I=<x³-1, 2x⁴+2x³+7x²+5x+ 5>in Q[x]
Find
p(x) belongs to Q[x]such that I=<p(x)> .
Is Q[x]/I is field ? Give reasons for your answer
1
Expert's answer
2020-03-16T14:51:55-0400

"p(x)=(x^3-1,2x^4+2x^3+7x^2+5x+5)" is the greatest common divisor of 

"x^3-1" and "2x^4+2x^3+7x^2+5x+5" .

Find "p(x)" according to Euclid's algorithm

"2x^4+2x^3+7x^2+5x+5:x^3-1"


"\\begin{matrix} 2x^4+2x^3+7x^2+5x+5 &&&& |x^3-1 \\\\ \n2x^4-2x &&&& 2x+2\\\\ \n&&2x^3+7x^2+5x+5\\\\\n &&2x^3-2&\\\\\n &&&7x^2+7x+7\\\\ \n&&&x^2+x+1 \\end{matrix}"


"x^3-1:x^2+x+1"


"\\begin{matrix} x^3-1 &&& |x^2+x+1 \\\\ x^3+x^2+x &&&x-1 \\\\ & -x^2-x-1\\\\ &-x^2-x-1\\\\ &&0 \\end{matrix}"


Then "p(x)=x^2+x+1" .


"Q[x]\/p(x)=Q[x] \/<x^2+x+1>" .


The set "Q[x]\/p(x)"  the product of polynomials from "Q[x]" by "p(x)=x^2+x+1" .

We are obsessed with polynomials "Q[x]" .

Then "Q[x]\/ p(x)=Q[x]\/Q[x]=\\{0\\}"


It is not a field, because there  "x^2+x+1=0"


Answer: "p(x)=x^2+x+1" , "Q[x]\/I" is not a field


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Comments

Assignment Expert
12.10.20, 21:59

Dear Ram, please use the panel for submitting new questions.

Ram
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Assignment Expert
08.10.20, 17:58

Dear Ram, please use the panel for submitting new questions.

Ram
08.10.20, 16:52

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